Correlation and Regression - textbook p304 eg17 explanation

[pablosdad]

Hi guys, I’m having difficulty understanding the logic and calculation of the explanations:

1. In this example on page 304, the textbook states “The value of the coefficient is only a little more than one-third the size of the standard error for that coefficient(0.0023), so the t-stat for the coefficient is only 0.4036”. I don’t know how this 0.4036 is arrived. My way is: (0.0009 - 0)/0.0023 = 0.3913.

2. The textbook explanation continues " Therefore , we cannot reject the null hypothesis (alpha =0) that the fund did not have a significant excess return beyond the return associated with the market risk of the fund." Does the textbook use a t-distribution chart value to compare and then make this conclusion? If so, doesn’t it need a level of significance? Or am I missing something here?

Thanks!

[Aether]

1. I think the t-stat is 0.3913, but the book has it as 0.4036 due to rounding. The output probably came from Excel. It’s conjecture on my part, but if you calcualte the beta t-stat, it is 18.95 (not 18.96 as the book says). I think since alpha is given as 0.0009, there was some info lost due to rounding.

2. The text does use a t distribution. I think in most cases you can use LOS = 5%. Since you have one variable, your df = 60 - 2 = 58. Since it’s a two-tailed test, your probabality in either tail is 0.025. The corresponding critical value is 2.000. Note that there is no df = 58 in the table… you interpolate to the closest df. Since 0.3913 (or 0.4036) is less than 2.000, you fail to reject the null. Meaning, the intercept is not significant. However, the slope (which is what we care about anyway) is highly significant (18.95 > 2.000). Ultimately, in this e.g., it doesn’t matter even if your LOS is as low as 1%. You’re still going to fail to reject the null for the intercept.

As a side note, when you get to the next chapter, you’ll find that a lot of formulas you learn in this chapter are generalized in reading 12. For example, in simple linear regression, they simply tell you that the degrees of freedom is (n - 2). But the general formula is (n - k - 1), where k is the number of independent variables. It’s just simpler to remember (n - k - 1) because the formula also works for simple linear regression with one variable. So don’t spend too much time memorizing formulas from this chapter.

Hope this helps.

[pablosdad]

Thanks for the explanation and headsup for chapter 12. All clear now

@Aether

1. I think the t-stat is 0.3913, but the book has it as 0.4036 due to rounding. The output probably came from Excel. It’s conjecture on my part, but if you calcualte the beta t-stat, it is 18.95 (not 18.96 as the book says). I think since alpha is given as 0.0009, there was some info lost due to rounding.

2. The text does use a t distribution. I think in most cases you can use LOS = 5%. Since you have one variable, your df = 60 - 2 = 58. Since it’s a two-tailed test, your probabality in either tail is 0.025. The corresponding critical value is 2.000. Note that there is no df = 58 in the table… you interpolate to the closest df. Since 0.3913 (or 0.4036) is less than 2.000, you fail to reject the null. Meaning, the intercept is not significant. However, the slope (which is what we care about anyway) is highly significant (18.95 > 2.000). Ultimately, in this e.g., it doesn’t matter even if your LOS is as low as 1%. You’re still going to fail to reject the null for the intercept.

As a side note, when you get to the next chapter, you’ll find that a lot of formulas you learn in this chapter are generalized in reading 12. For example, in simple linear regression, they simply tell you that the degrees of freedom is (n - 2). But the general formula is (n - k - 1), where k is the number of independent variables. It’s just simpler to remember (n - k - 1) because the formula also works for simple linear regression with one variable. So don’t spend too much time memorizing formulas from this chapter.

Hope this helps.